Theorem cnmpt1vsca 18223

Description: Continuity of scalar multiplication; analogue of cnmpt12f 17698 which cannot be used directly because .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
tlmtrg.f	|- F = (Scalar ` W )
cnmpt1vsca.t	|- .x. = ( .s ` W )
cnmpt1vsca.j	|- J = ( TopOpen ` W )
cnmpt1vsca.k	|- K = ( TopOpen ` F )
cnmpt1vsca.w	|- ( ph -> W e. TopMod )
cnmpt1vsca.l	|- ( ph -> L e. (TopOn ` X ) )
cnmpt1vsca.a	|- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
cnmpt1vsca.b	|- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )

Assertion

Ref	Expression
cnmpt1vsca	|- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Distinct variable groups:   x, F   x, J   x, K   x, L   ph, x   x, W   x, X

Allowed substitution hints:   A( x)   B( x)   .x. ( x)

Proof of Theorem cnmpt1vsca

Step	Hyp	Ref	Expression
1		cnmpt1vsca.l	. . . . . . 7 |- ( ph -> L e. (TopOn ` X ) )
2		cnmpt1vsca.w	. . . . . . . . 9 |- ( ph -> W e. TopMod )
3		tlmtrg.f	. . . . . . . . . 10 |- F = (Scalar ` W )
4	3	tlmscatps 18220	. . . . . . . . 9 |- ( W e. TopMod -> F e. TopSp )
5	2, 4	syl 16	. . . . . . . 8 |- ( ph -> F e. TopSp )
6		eqid 2436	. . . . . . . . 9 |- ( Base ` F ) = ( Base ` F )
7		cnmpt1vsca.k	. . . . . . . . 9 |- K = ( TopOpen ` F )
8	6, 7	istps 17001	. . . . . . . 8 |- ( F e. TopSp <-> K e. (TopOn ` ( Base ` F ) ) )
9	5, 8	sylib 189	. . . . . . 7 |- ( ph -> K e. (TopOn ` ( Base ` F ) ) )
10		cnmpt1vsca.a	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
11		cnf2 17313	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ K e. (TopOn ` ( Base ` F ) ) /\ ( x e. X |-> A ) e. ( L Cn K ) ) -> ( x e. X |-> A ) : X --> ( Base ` F ) )
12	1, 9, 10, 11	syl3anc 1184	. . . . . 6 |- ( ph -> ( x e. X |-> A ) : X --> ( Base ` F ) )
13		eqid 2436	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
14	13	fmpt 5890	. . . . . 6 |- ( A. x e. X A e. ( Base ` F ) <-> ( x e. X |-> A ) : X --> ( Base ` F ) )
15	12, 14	sylibr 204	. . . . 5 |- ( ph -> A. x e. X A e. ( Base ` F ) )
16	15	r19.21bi 2804	. . . 4 |- ( ( ph /\ x e. X ) -> A e. ( Base ` F ) )
17		tlmtps 18217	. . . . . . . . 9 |- ( W e. TopMod -> W e. TopSp )
18	2, 17	syl 16	. . . . . . . 8 |- ( ph -> W e. TopSp )
19		eqid 2436	. . . . . . . . 9 |- ( Base ` W ) = ( Base ` W )
20		cnmpt1vsca.j	. . . . . . . . 9 |- J = ( TopOpen ` W )
21	19, 20	istps 17001	. . . . . . . 8 |- ( W e. TopSp <-> J e. (TopOn ` ( Base ` W ) ) )
22	18, 21	sylib 189	. . . . . . 7 |- ( ph -> J e. (TopOn ` ( Base ` W ) ) )
23		cnmpt1vsca.b	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )
24		cnf2 17313	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ J e. (TopOn ` ( Base ` W ) ) /\ ( x e. X |-> B ) e. ( L Cn J ) ) -> ( x e. X |-> B ) : X --> ( Base ` W ) )
25	1, 22, 23, 24	syl3anc 1184	. . . . . 6 |- ( ph -> ( x e. X |-> B ) : X --> ( Base ` W ) )
26		eqid 2436	. . . . . . 7 |- ( x e. X |-> B ) = ( x e. X |-> B )
27	26	fmpt 5890	. . . . . 6 |- ( A. x e. X B e. ( Base ` W ) <-> ( x e. X |-> B ) : X --> ( Base ` W ) )
28	25, 27	sylibr 204	. . . . 5 |- ( ph -> A. x e. X B e. ( Base ` W ) )
29	28	r19.21bi 2804	. . . 4 |- ( ( ph /\ x e. X ) -> B e. ( Base ` W ) )
30		eqid 2436	. . . . 5 |- ( .s f ` W ) = ( .s f ` W )
31		cnmpt1vsca.t	. . . . 5 |- .x. = ( .s ` W )
32	19, 3, 6, 30, 31	scafval 15969	. . . 4 |- ( ( A e. ( Base ` F ) /\ B e. ( Base ` W ) ) -> ( A ( .s f ` W ) B ) = ( A .x. B ) )
33	16, 29, 32	syl2anc 643	. . 3 |- ( ( ph /\ x e. X ) -> ( A ( .s f ` W ) B ) = ( A .x. B ) )
34	33	mpteq2dva 4295	. 2 |- ( ph -> ( x e. X |-> ( A ( .s f ` W ) B ) ) = ( x e. X |-> ( A .x. B ) ) )
35	30, 20, 3, 7	vscacn 18215	. . . 4 |- ( W e. TopMod -> ( .s f ` W ) e. ( ( K tX J ) Cn J ) )
36	2, 35	syl 16	. . 3 |- ( ph -> ( .s f ` W ) e. ( ( K tX J ) Cn J ) )
37	1, 10, 23, 36	cnmpt12f 17698	. 2 |- ( ph -> ( x e. X |-> ( A ( .s f ` W ) B ) ) e. ( L Cn J ) )
38	34, 37	eqeltrrd 2511	1 |- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1652   e. wcel 1725   A.wral 2705   e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   TopOpenctopn 13649   .s fcscaf 15951  TopOnctopon 16959   TopSpctps 16961   Cn ccn 17288   tX ctx 17592  TopModctlm 18187

This theorem is referenced by:  tlmtgp  18225

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-slot 13473  df-base 13474  df-topgen 13667  df-scaf 15953  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cn 17291  df-tx 17594  df-tmd 18102  df-tgp 18103  df-trg 18189  df-tlm 18191